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THE 



SQUARE ROOT OF SURDS: 
SOLUTION- 



m 



XLVII. PROBLEM 



EUKGMl 



SQUARE OF THE CIRCLE, 



WITH THE TRUE METHOD OF FINDING THE 



CIRCUMFERENCE. 



DISCOVERED BY 

D. S. MERCERON, 

BALTIMORE, 

1847. 



PRINTED BY SAMUEL SANDS, BALTIMORE. 




mm 




THE 



SQUARE ROOT OF SURDS:: 
SOLUTION 

OF THE 

XLVII. PROBLEM 



SQUAKE OF THE CIRCLE, 

WITH THE TRUE METHOD OF FINDING THE 



CIRCUMFERENCE. 



DISCOVERED BY 

D. S. MERCERON, 

BALTIMORE, 

1847. 



TRINTED BY SAMUEL SANDS, BALTIMORE, 

1848. 



Entered according to the Act of Congress, in the year 1848, by 

D. S. MERCERON, 

In the Clerk's Office of the District Court of the United States, 
in and for the State of Maryland. 






PRELIMINARY REMARK. 



This System is founded in Vulgar Fractions, by which, and 
by which alone, the proper results can be obtained; and any at- 
tempt to work by Decimals, either in whole or in part, will prove 
abortive ; for it is so constructed as in the first operation, viz : 
" the Square of the Root" to exhibit on every occasion, whether 
the term of the fraction be high or low, the remainder in excess 
of 1. But the construction of this System is of such an extraor- 
dinary character as to exhibit in the course of its several opera- 
tions, a property in the combination of numbers hitherto unknown, 
viz : that of PURGING AND RIDDING ITSELF OF THE 
EXCRESCENCE ENGRAFTED UPON IT, AND COMING 
OUT IN THE END PERFECTLY TRUE AND WITHOUT 
ANY REMAINDER WHATEVER ; just like old metal, when 
put into a crucible, after undergoing the operation of fusion, the 
pure part is collected at the bottom and the dross thrown off. 



TABLE 



SQUARE-ROOT OF SURDS; 

By which the Root of any Surd ivhole No. or of any Surd fraction may 

be obtained. 



SQUARE-R 


OOT. 


Sq. No. 


SQUARE-ROOT. 


Sq. No. 


Integer 


Numerator. 


Denominator 




Integer 

34 


Numerator. 


Denominator. 




1 


11 


15 


3 


4829 


4899 


1224 


2 


29 


35 


8 


35 


5111 


5183 


1295 


3 


55 


63 


15 


36 


5401 


5475 


1368 


4 


89 


99 


24 


37 


5699 


5775 


1443 


5 


131 


143 


35 


38 


6005 


6083 


1520 


6 


181 


195 


48 


39 


6319 


6399 


1599 


7 


239 


255 


63 


40 


6641 


6723 


1680 


8 


305 


323 


80 


41 


6971 


7055 


1763 


9 


379 


399 


99 


42 


7309 


7395 


1848 


10 


461 


483 


120 


43 


7655 


7743 


1935 


11 


551 


575 


143 


44 


8009 


8099 


2024 


12 


649 


675 


168 


45 


8371 


8463 


2115 


13 


755 


783 


195 


46 


8741 


8835 


2208 


14 


869 


899 


224 


47 


9119 


9215 


2303 


15 


991 


1023 


255 


48 


9505 


9603 


2400 


16 


1121 


1155 


288 


49 


9899 


9999 


2499 


17 


1259 


1295 


323 


50 


10301 


10403 


2600 


18 


1405 


1443 


360 


51 


10711 


10815 


2703 


19 


1559 


1599 


399 


52 


11129 


11235 


2808 


20 


1721 


1763 


440 


53 


11555 


11663 


2915 


21 


1891 


1935 


483 


54 


11989 


12099 


3024 


22 


2069 


2115 


528 


55 


12431 


12543 


3135 


23 


2255 


2303 


575 


56 


128S1 


12995 


3248 


24 


2449 


2499 


624 


57 


13339 


13455 


3363 


25 


2651 


2703 


675 


58 


13805 


13923 


3480 


26 


2861 


2915 


7'28 


59 


14279 


14399 


3599 


27 


3079 


3135 


783 


6.0 


14761 


14883 


3720 


28 


3305 


3363 


840 


61 


15251 


15375 


3843 


• 29 


3539 


3599 


899 


62 


15749 


15875 


3968 


30 


3781 


3843 


960 


63 


16255 


16383 


4095 


31 


4031 


4095 


1023 


64 


16769 


16899 


4224 


32 


4289 


4355 


1088 


65 


17291 


17423 


4355 


33 


4555 


4623 


1155 


66 


17821 


17955 


4488 



TABLE CONTINUED. 



SQUARE-R 


OOT. 


Sq. No. 


sc^uare-roOt. 


Sq. No. 


Integer. 


Numerator. 


Denominator 




Integer. 


Numerator. 


Denominator. 




67 


18359 


18495 


4623 


88 


31505 


31683 


7920 


68 


18905 


19043 


4760 


89 


32219 


32399 


8099 


69 


19459 


19599 


4899 


90 


32941 


33123 


8280 


70 


20021 


20163 


5040 


91 


33671 


33855 


8463 


71 


20591 


20735 


5183 


92 


34409 


34595 


8648 


72 


21169 


21315 


5328 


93 


35155 


35343 


8835 


73 


21755 


21903 


5475 


94 


N 35909 


36099 


9024 


74 


22349 


22499 


5624 


95 


36671 


36863 


9215 


75 


22951 


23103 


5775 


96 


37441 


37635 


9408 


76 


23561 


23715 


5928 


97 


38219 


38415 


9603 


77 


24179 


24335 


6083 


98 


39005 


39203 


9800 


78 


24805 


24963 


6240 


99 


39799 


39999 


9999 


79 


25439 


25599 


6309 


100 


40601 


40803 


10200 


80 
81 


26081 
26731 


26243 
26895 


6560 
6723 


1 


810 


812 


203 










82 


27389 


27555 


6888 





99 


140 


n 


83 


28055 


28223 


7055 


19601 


27720 


f i 


84 


28729 


28899 


7224 




3880899 


5488420 


i 2 


85 


29411 


29583 


7395 




3650401 


6322680 


\ 


86 


30101 


30275 


7568 




3650401 


4215120 


3 


87 


30799 


30975 


7743 









RULE POR CONTINUING THE TABLE. 
To the column of Integers add 1 each time. 

To the column of Numerators add 810 for the first, and an additional 
8 each time after. 

To the column of Denominators add 812, and an additional 8 each time 
after — and to the column of Square Nos. add 203 and an additional 2 each 
time after. 



RULE 

For Extracting the Roots of Surds. 

1st — By Division. 

Select from the Table any Surd which when divided by a square num- 
ber or a series of square numbers will give the required No. Then divide 
the Root of the No. so selected by the Root of the Square No. by which 
it has been divided and the quotient will be the Root of the required Surd. 



6 

Example. 

Required to find the Roots of the Surds, 

72—18—41—32—8—2 and }, 

Select the Surd 288 the Root of which is 16—1121—1155. 

4)288 2)16 1121 1155 

2 



4)72 


2)8 


u 


2310 
2 


4)18 


2)4 


u 


4620 
2 


4* 


2 

Again : 


u 


9240 


9)288 

4)32 

4)8 

4)2 


3)16 

2)5 

2)2 

2)1 




1121 
2276 
5741 

u 

19601 


1155 

3465 

6930 

13860 

27720 



2d — By Multiplication. 

Select from the table any Surd which when multiplied by a square num- 
ber will give the one required ; then multiply the Root of the selected 
number by the Root of the number by which it is multiplied, and the pro- 
duct will be the Root of the number sought. 

Example. 

The Root of the Surd 7 being found, it is required to find the Root of 
the Surd 28. 

7 2 494 765 

2 



4 



765 



28 5 223 



But as the value of the remainder 1 which runs through the whole sys- 
tem is by the operation of multiplication increased (contrary to that of di- 
vision by which it is decreased) it is advisable before proceeding to operate 
by this Rule to reduce the value of the remainder 1, which can be done to 
infinity by the following operations. 

RULE 

For Reducing the value of the remainder 1. 
1st Operation. — Multiply the Numerator by 2, and the Denominator by 
twice the integer — add the two products' together for a common multiplier 



— then multiply the Numerator and Denominator by that common multi- 
plier for a new Numerator and Denominator, taking care to subtract 1 
from the new Numerator alone. 

Example. 
Required to reduce the value of the remainder 1 from the Square of the 
Root of the Surd 8. 

Surd 8— Root— 2 29 35 1st term 

2 4 



58 



Deduct, 



198 
29 

5742 
1 



140 

58 



198 common multiplier. 
35 



New Numerator, 5741 6930 New Denominator. 

2d — And all future operations. — Multiply the 2d or new term, by the 
common multiplier, from the product of which subtract the 1st term and 
the remainders will be the 3d term. 

Again — Multiply the 3d term by the common multiplier, from the pro- 
duct of which subtract the 2d term, and the remainders will be the 4th 
term ; and so on to infinity. 

Example. 

2d term. 



Subtract 



Subtract 



5741 
198 


6930 
198 


1136718 
29 


1372140 
35 


1136689 
198 


1372105 
198 


225064422 
5741 


271676790 
6930 


225058681 


271669S60 



3d term. 



4th term. 



RULE 

For transferring the remainder 1 from the Square of the Numerator (its 

natural place) to the Square of the Denominator of Roots consisting 

of simple fractions. 

Divide the Denominator by the Square of the fraction, and place the 
quotient over the Numerator and the result will be the fraction required. 
The quotient of the Denominator thus becoming the Numerator, and the 
original Numerator becoming the Denominator, 



8 



J 


Example. 

70 13860 2744210 

99—i)140 — 19601—|)27720 = 3S80899— =1)5488420 

1st Term. 2d Term. 3d Term. 
2107560 


I 


3650401—^)6322680. 
3161340 




1 

The 


3650401—|)4215120 

square of ( 2744210 — 
« i ? 3880899 — 
« ^5488420 — 


Proof. 
7530688524100 
15061377048201 
30122754096400 



Note. — In squaring the hypothenuse of the Right-angled Triangle, the 
remainder 1 from the facility afforded as above of transferring it from plus 
to minus et vice versa, ought to be expunged, and the substituted in its 
place ; for when the 1 is so transferred the deficit is exactly the same as 
the former excess ; and if both terms are operated the one neutralizes the 
other. 



S © IL w IF H © H 



XLVII. PROBLEM 



OP 



EUCLID. 



Viz : — That the Square of the hypothenuse of a Right-angled Triangle is 
exactly equal to the squares of the other two sides united. 








AREA 
4 


to %? 




2 \ 




AREA 




4 



Note 1. — The hypothenuse of the Right-angled Triangle, and the chord 
of the arc of 90° of a circle are one and the same. 
2 



10 



SWAMM ©IF TMM (0m<SSMo 

RULE 

For Squaring the Circle. 
Multiply the chord of the arc of 90° by 5, and divide the product by 4, 
and the quotient will be the circular Square-Root, which, when multiplied 
will give the square area of the Circle. 

of 12. 



Example 

Chord of the Arc of 90° is, 
Multiplied by 5 = 
Divided by 4 = 

The square of which is 



ON A DIAMETER 

8 — 2663428 — 5488420 
42 — 2340300 — " 
10—3329285— « 
112 1 225 -^ 

C 30122754096400 
Excess created by the remainder 1, { 1 

( 133S78907095 



^ 



1 



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s 



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v. 



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o 



^ 






& 





a 
o 

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63 


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00 
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r— 1 


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MNi-iMSCD Mb- OCOb-H»f3 
^GQ^iritDGOOiOrHCQ'^iO 




fc. 

a 
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o 


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o 


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5 


c 

03 

Q 






s-, 


SO 


t-h ^ OS O tft O 
-^ CM CO 




'$ 


^o 


The square of every No. is here exhibited. 




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01 piTB J01T2UIOIOU9Q oi pguoryodoid SS90Xg 


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11 

RULE 

For finding the Circumference of the Circle. 
Multiply the circular area of the diameter 1 by 4 ; then multiply the 
product by each diameter for its circumference. 

Example. 

Denom. 110880 



Circular Area of 1, as per table below, 86625 

4 



Circumference of 1 = 3.13860 

12 



12 — 37.55440 



Note 1. — The chord of the Arc of 90° is composed of the circumfer- 
ence and diameter, united and blended in one straight line in true and equal 
proportions, and is therefore, the only true principle upon which the Circle 
can be squared, or the circnmference found; and every other segment of the 
Circle is inaccurate as containing undue proportions. 

2. — The multiplier and divisor 5 and 4 seem to have been designed by 
the Great Mathematician of the Universe for squaring the Circle by, be- 
cause they leave no remainder. The excess which appears being created 
by the universal 1 upon which the whole fabric is built, and which is es- 
sential to the different operations, but which is ultimately thrown off and 
rejected as valueless. 



OBSERVATIONS. 



1. The foregoing Table of the Square of the Circle is formed from the 
2d term of the Root J, and the previous example taken from the 3d term, 
in order to show how materially the excess created by the remainder 1 is 
reduced by only one remove : the Table giving on the diameter 12 an excess 
°f TT29 S 4T7T4tfTr = 34T1104 an d the example giving on the same diam- 
eter an excess of only OTwWnrrani = twhtwiv* 

2. This system is built not only mathematically but geometrically on the 
Square, having 1 for its base or corner stone, as may be perceived by a 
glance at the 

KEY 
Of the Square Root of Surds, the foundation of the system. 
12 4 3 



1st. Numerator 11 8 8 15 1st Denominat. 

But the Great architect of the Universe who formed the system, and who 

has no need of any base or foundation whereon to rest His superstructures, 



VI 

has, after forming this system, cast away and rejected the corner stone as 
being no longer necessary to support the fabric — just as the human archi- 
tect finds it necessary in the construction of an arch to build it on a frame, 
but which, when completed, is not only useless but an obstruction. 

3. The great obstacle which has hitherto prevented mathematicians from 
discovering this system, is the universal practice adopted by them, in making 
use of Decimals instead of Vulgar fractions, for the decimal is a fraction- of 
an even number, whereas this system is entirely composed of odd numbers, 
both numerator and denominator, as a glance at the Table of Roots will 
show. 

4 As a proof of the superiority of vulgar fractions over decimals, it is 
only necessary to produce the following illustration : Take, for instance, 
the Root of J, which is 3880899—5488420 ; convert it into a decimal 
having the same number of figures as the vulgar fractions, viz : 7 ; the de- 
cimal will then be .7071067. Now the vulgar fraction when squared pro- 
duces ^|0|^3 7jo4||.oi gi ving an excess of only TTrTirTr J, W5TTrTr and the 
Decimal when squared produces .49999988518489, leaving a deficiency of 
.00000011481511, equal to ■§yt>16s£ an enormous difference, and yet both 
produced from the same fraction. 

5 What is very remarkable and helps to shew the perfection of this system 
is the manner in which the excess created by the remainder 1, is in the final 
operation of squaring the circle separated and distinguished from the true 
result, (which leaves no remainder whatever) so that it can never be mista- 
ken ; for it is as distinctly marked as the scum thrown up in a cauldron of 
boiling liquid and collected on one side. 

6. Another proof of the perfection of this system is evinced in the extrac- 
tion of the Root of the same Surd from different parts of the Table, (which 
often occurs) for altho' each Root is different in its numbers, yet their val- 
ues are all equal (except as to that of the rem. 1) and lead to one result: 
indeed the whole system is throughout so dove-tailed and interwoven that 
it forms a perfect whole. As an example of which, it is only necessary to 
extract the Root of \ from the Roots of the Surds 8—288 and 9800. 

4)8 2)2 29 35 



4) 2 2) 1 70 

4 99 140 



288 16 1121 1155 



This example is shown before, where the Root of 
J is 19601 27720 






13 



100)9800 


10)98 

7)9 

2)1 




39005 


39203 


49)98 


352629 


392030 


4J)2 


1136689 


2744210 


_J 


3880899 


5488420 



7. A remarkable feature of this system, and which helps to shew its per- 
fection, is exhibited in the Table of Roots where the 1st Denominator be- 
comes the 3d square No., the 2d Denominator becomes the 5th square No. 
the 3d Denominator becomes the 7th square No., and so on throughout ; 
the square No. outstripping the Denominator in the ratio of 2 to 1. It is 
also worthy of remark, that the last figure of each No. both of the Nume- 
rator and Denominator is repeated in a regular series of 5 each, consisting 
of 3 figures, viz : 1 — 5 — 9 for the Numerator, and 3 — 5 — 9 for the De- 
nominator, the 1 and 9 being twice used in the Numerator, and the 3 aud 5 
twice in the Denominator, but neither of them ever employing the 7 which 
together with all the even figures are carefully excluded. 

8. The whole system of squaring the circle and finding the value of its 
circumference may be comprized in these few words : The Root of J de- 
termines the value of the Chord of 90° — the Chord of 90° produces the 
circular square Root — the circular square Root produces the area and the 
area gives the value of the circumference, thus working its way from the 
centre to the circumference. The remainder 1, which may be regarded as 
the centre, may be traced from the Root of the Surd, where it originates, 
through each operation until it arrives at the area, which is circumscribed 
by the circumference, where it is finally purged and cast off, but which is 
as necessary in the different operations as the leaven to the bread and the 
lees to the wine. 

9. In viewing the result obtained by this system, and comparing it with 
what has been hitherto laid down in all mathematical works as the con- 
tents of the Circle, a very considerable difference is manifested, amounting 
in a circular foot to half a square inch and a fraction. This excess over 
the true quantity arises from several causes : the first and principal of 
which has been the want of the Surd Root, whereby mathematicians have 
been compelled to work all round the Circumference instead of beginning 
from the centre : the next cause is, that from the same want of the Surd 
Root, the true circumference has never been known, and they have there- 
fore worked in the dark as far as that knowledge was wanting, the cir- 
cumference being the last thing obtained ; thus beginning at the roof in- 
stead of the foundation: the next cause is that in working round the cir- 
cumference they have employed segments of less than 90°, all of which 
contain a greater proportion of the circumference than of the diameter, 
which has consequently swelled the area beyond its true quantity and this 
excess is increased in proportion as the segment is decreased and vice versa : 
Another cause is to be found in the use of decimals instead of vulgar frac- 
tions, from the fact that a decimal, as shewn in the body of this work, 
leaves a much greater remainder when squared than the vulgar fraction ; 
for being the fraction of an even number, it can never be made to approxi- 
mate so closely as an odd one allows. 

DANIEL S. MERCERON. 

Baltimore, 17 th January, 1848. 




3*? <?< 




GRAND DISCOVERT 



IN 



MATHEMATICS ! 



Wonderful Propei ty in the Combination of Nos ! 



Just Published, a work entitled " The Square Root of 
Surds : Solution of the XLVII. Problem of Euclid, and 
Square of the Circle, with the true method of finding the 
Circumference." 

This work is the production of D. S. Merceron, of Balti- 
more, who has labored at it for more than two years, and 
solves that intricate question which for more than 2000 
years has puzzled the brain of all the Mathematicians in 
the World, viz: THE SQUARE OF THE CIRCLE, 
which it does completely and effectually, without leaving 
the slightest remainder: thus proving that by patience, 
energy and perseverance, nothing is impossible to the Mind 
of Man, and that whatever he wills he can accomplish. 

It is published in a neat octavo form, with tables and 
diagrams, and proper rules and examples for working, and 
may be had of 

No. 2, Jarvis' Building, Baltimore, Md. 
Price 12| cents, or 10 copies for 81. 










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